By Jae K. Lee

ISBN-10: 0471692727

ISBN-13: 9780471692720

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Suppose that a system is tested for a fault once every hour. 05. (a) What is the probability that a fault will be detected in exactly 3 hours? (b) In 3 hours or more? (c) In more than 3 hours, given that it has not been detected for 2 hours? (d) What is the average time to detection of a fault? Solution Let X be the time to detection of a fault. 95. 0024. (b) P(X ! 3) ¼ P(X . 0025. 95) (c) P(X . 3 j x . 2) ¼ P(X . 05. 053 hours. 5 Negative Binomial Distribution The binomial distribution counts the number of successes in a ﬁxed number of Bernoulli trials, each with a probability p of success.

What is the distribution of the sum? Solution Let X be the sum of the numbers on two fair tossed dice. f. of X can be derived by listing all 36 possible outcomes, which are equally likely, and counting the outcomes that result in X ¼ x for x ¼ 2, 3, . . , 12. f. 1 Histogram of X, the sum of the numbers on two fair dice. 5 EXPECTED VALUE AND VARIANCE 15 For example, there are 4 outcomes that result in X ¼ 5: (1, 4), (2, 3), (3, 2), (4, 1). f. 1. v. X is continuous if it can take any value from one or more intervals of real numbers.

The mean and variance of X are given by E(X) ¼ 1 l and Var(X) ¼ 1 l2 Since 1/l is the average time between two events, l can be interpreted as the rate at which events occur. It can be shown that if events occur according to a Poisson process with rate l, then the interevent times are exponentially distributed with mean 1/l and vice versa. f. of the exponential distribution with fX (x) ¼ 2eÀ2x : Memoryless Property The exponential distribution shares the memoryless property with the geometric distribution, and it is the only continuous distribution having this property.